Moduli spaces in $p$-adic non-abelian Hodge theory

Abstract

We propose a new moduli-theoretic approach to the $p$-adic Simpson correspondence for a smooth proper rigid space $X$ over ${\mathbb{C}}_p$ with coefficients in any rigid analytic group $G$, in terms of a comparison of moduli stacks. For its formulation, we introduce the class of "smoothoid spaces" which are perfectoid families of smooth rigid spaces, well-suited for studying relative $p$-adic Hodge theory. For any smoothoid space $Y$, we then construct a "sheafified non-abelian Hodge correspondence", namely a canonical isomorphism \[R^1ν_{\ast}G\xrightarrow{\sim} \mathrm{Higgs}_G\] where $\nu :Y_v\to Y_{et}$ is the natural morphism of sites, and where ${\mathrm{Higgs}}_G$ is the sheaf of isomorphism classes of $G$-Higgs bundles on $Y_{et}$. We also prove a generalisation of Faltings' local $p$-adic Simpson correspondence to $G$-bundles and to perfectoid families. We apply these results to deduce $v$-descent criteria for étale $G$-bundles which show that $G$-Higgs bundles on $X$ form a small $v$-stack $\mathcal{H}igg{s}_G$. As a second application, we construct an analogue of the Hitchin morphism on the Betti side: a morphism $\mathcal{B}u{n}_{G,v}\to {\mathcal{A}}_G$ from the small $v$-stack of $v$-topological $G$-bundles on $X$ to the Hitchin base. This allows us to give a conjectural reformulation of the $p$-adic Simpson correspondence for $X$ in a more geometric and more canonical way, namely in terms of a comparison of Hitchin morphisms.

updated now that sequel has appeared and some small corrections

Cited by in corpus (1)

• The small $p$-adic Simpson correspondence in terms of moduli spaces